Separable fuzzy soft sets and decision making with ...

Accepted Manuscript Title: Separable fuzzy soft sets and decision making with positive and negative attributes Author: Jos´e Carlos R. Alcantud Terry Jacob Mathew PII: DOI: Reference:

S1568-4946(17)30353-8 http://dx.doi.org/doi:10.1016/j.asoc.2017.06.010 ASOC 4279

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

12-10-2016 31-5-2017 5-6-2017

Please cite this article as: Jos´e Carlos R. Alcantud, Terry Jacob Mathew, Separable fuzzy soft sets and decision making with positive and negative attributes, (2017), http://dx.doi.org/10.1016/j.asoc.2017.06.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights (for review)

Highlights • The novel notion of separable fuzzy soft set is presented

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• Normalizes positive and negative attributes by application of fuzzy complements

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• Introduces a new sensitivity concept to discriminate among decision making procedures

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• An innovative aggregation process by parameter classes to obtain H-resultant fuzzy soft set

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• A new flexible algorithm with high discrimination to find the optimal alternative is proposed

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Graphical abstract (for review)

Mix of +ve and –ve parameters fss

Normalized by fuzzy complements

Fuzzy soft set

Only +ve parameters (F,A)

fss

(F,A {A1, A2,A3}) - separable fuzzy soft sets

Partitioned to

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(F,A)

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(F,A)

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Separable fuzzy soft set decision making with mixed parameters

(F1,A) (F2,B) (F3,C) A1

A2

A3

Aggregation function

H-resultant fuzzy soft set

DM algorithm

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Manuscript PDF format Click here to view linked References

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Separable fuzzy soft sets and decision making with positive and negative attributes Jos´e Carlos R. Alcantuda,∗, Terry Jacob Mathewb a

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BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, 37007 Salamanca, Spain [email protected], URL: http://diarium.usal.es/jcr b School of Computer Sciences, Mahatma Gandhi University, Kottayam, Kerala, India [email protected]

Abstract

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A novel notion of separable fuzzy soft set is presented. In this context, the problem of decision making in the presence of possibly negative attributes is studied. This solution introduces the following two innovations: First, a fuzzy complement enables the information to be uniform when there are attributes that the decision maker regards as negative. Such a possibility defines the new notion of sensitivity of decision making procedures for fuzzy soft sets to the choice of a fuzzy complement. Our sensitivity concept introduces a new criterion to discriminate among decision making procedures for fuzzy soft sets. Secondly, an aggregation process by parameter classes produce a resultant fuzzy soft set. The mechanism that decides among the alternatives in this resultant fuzzy soft set rests on scores computed from a relative comparison matrix. In this way, we obtain an adjustable solution to the problem of decision making for separable fuzzy soft sets. This flexible novel procedure achieves a high power of discrimination and produces a welldetermined optimal alternative. Our model is validated through a real case study. Keywords: Fuzzy soft set; Separable fuzzy soft set; Comparison table; Fuzzy complement; Decision making.

∗

Corresponding author

Preprint submitted to Applied Soft Computing

May 30, 2017

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1. Introduction

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A large number of problems in this application-oriented real world rely on imprecise, subjective and uncertain data. Just because the data is inexact, the solutions too have to be modelled with mathematical tools that can handle uncertainty. Fuzzy sets and fuzzy logic were introduced by Zadeh [1] and since then, they have contributed to a paradigm shift in the way we deal with imprecise problems and their solutions. Fuzzy sets are highly successful in many areas and give better results than the classical approach. However, fuzzy sets are inferior in areas like setting the degree of membership, representing imprecise or loosely held data in practical situations, etc. Hence, to solve these problems, many researchers, such as Atanassov [2] put forward the concept of intuitionistic fuzzy sets; Pawlak [3] introduced rough sets; Torra [4] put forward the hesitant fuzzy sets, which are applicable in real-world situations as in Alcantud et al. [5], etc. However, these theories are limited due to the lack of parameterization concept associated with them for describing the problem. It is at this juncture that in 1999, Molodtsov [6] introduced soft sets and established the fundamental results of this theory. A soft set is a collection of approximate descriptions of an object and is used as a general mathematical tool for dealing with objects which have been defined using a very loose and hence very general set of characteristics. Ali et al. [7] and Feng and Li [8] also contributed to settle the fundamental laws that govern this notion. Extensions and hybrid models that combine the soft set model with others have been defined and used for decision making e.g., in Ali [9], Das [10], Feng [11], Feng et al. [12, 13], Hakim et al. [14], Ma et al. [15], Peng and Yang [16], Zhan and Davvaz [17], Zhan et al. [18] and Zhu and Zhan [19]. Association rules mining have also benefitted from soft set theory, as in Feng et al. [20] and Herawan and Deris [21]. The absence of any form of restrictions in describing the problem space make soft set theory more appealing and flexible for application. Moreover, there is flexibility in choosing any form of parameterization, such as words and sentences, real numbers, functions, mappings and so on. It means that the problem of setting the membership function or any similar problems do not arise in soft sets. The motivation of this paper lies in the fact that the selection of an object based on a group of like minded parameters gives better decisions and solutions than the selection made from a combination of mixed group or unlikely 2

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group of parameters. Nevertheless, the parameter options are often not on a uniform scale. They may be subjectively pleasant or unpleasant in nature, i.e., positive or negative. A methodology to transform the different deciding options or factors for a decision can bring out better discrimination among the decisions made in the decision making process. By this selective separation of options and transformation to a uniform scale, we achieve modularity and hence, the Separation of Concerns relating to the choice of an optimal solution. Our study having been motivated by these concerns, we propose and investigate the novel concept of a separable fuzzy soft set. An adjustable algorithm for its decision making is proposed. It rests on two inputs which in special cases can be dispensed with, namely, a fuzzy complement and an interclass aggregation function. They are applied to the decision maker’s classification of the parameters and her judgement about which are negative. The organization of this paper is as follows. Section 2 recalls the basic definitions and makes a thorough inspection of algorithms contributed by Roy and Maji and Alcantud. Then comes the interesting case of decision making in the light of positive and negative attributes. The uncharted problem of negative attributes and their significance are highlighted in subsection 2.2. Here, we resort to fuzzy complements as the agent for normalizing the positive and negative attributes. This is done with the help of the popular Sugeno and Yager classes of fuzzy complements (cf., Klir and Yuan [22, Sect. 3.2]). We go on to thoroughly investigate, and compare Roy and Maji’s [23] algorithm with Alcantud’s algorithm [24] on the basis of the changes made in the ordering of objects under consideration. Thus by analysing the impact of fuzzy complements on these algorithms, we define the notion of sensitivity for a decision making procedure. In section 3 we investigate the novel notion of separable fuzzy soft set. The formal definition of separable fuzzy soft set is provided and its modularized tabular representation is illustrated. The transition to H-resultant fuzzy soft set is shown in section 3.2. We propose the flexible algorithm of decision making for a separable soft set with preset requirements in section 3.3. Section 3.4 gives additional remarks on our proposal. Section 3.5 validates the model with a real case study. We conclude in section 4.

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2. Fuzzy soft set decision making with positive and negative attributes

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2.1. Definitions: soft set and fuzzy soft set This section presents the basic definitions of soft set theory [6] and fuzzy set theory [25]. As usual, the common terminology for describing soft set and its extensions is followed. Here U refers to an initial universe and E is the set of parameters. Definition 1. (Molodtsov [6]) A pair (F, A) is a soft set over U when A ⊆ E and F : A −→ P(U ), where P(U ) denotes the set of all subsets of U .

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A soft set over U is regarded as a parameterized family of subsets of the universe U , the set A being the parameters. For each parameter e ∈ A, F (e) is the subset of U approximated by e or the set of e-approximate elements of the soft set. To put an example, if U = {c1 , c2 , c3 , c4 } is a universe of cars and A contains the parameter e that describes “family car”, and the parameter e0 that describes “sports car” then F (e) = {c1 } means that the only family car is c1 and F (e0 ) = {c2 , c3 } means that the cars coming under sports category are c2 and c3 . Many papers have conducted formal investigations of this and have worked on related concepts. For example, Maji, Biswas and Roy[26] developed the theoretical notion and defined soft subsets and supersets, soft equalities, intersections and unions of soft sets, et cetera. Feng and Li [8] give a systematic study on several types of soft subsets and various soft equal relations induced by them. Concerning exclusive soft set based decision making, we refer the reader to Maji et al.[27], C ¸ a˘gman and Engino˘glu [28] and Feng and Zhou [29]. In order to model more general situations, the following notion was subsequently proposed and investigated: Definition 2. (Maji et al.[25]) A pair (F, A) is a fuzzy soft set over U when A ⊆ E and F : A −→ FS(U ), where FS(U ) denotes the set of all fuzzy sets on U . Maji et al. [25] initiated the concept of the fuzzy soft set by blending the idea of fuzzy set [1]. By using this definition of fuzzy soft set many viable applications of soft set theory have been developed by researchers. Obviously, every soft set can be considered as a fuzzy soft set. Following on with our car example above, fuzzy soft sets enable us to deal with other 4

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properties, such as “fuel efficiency” or “stylish design”, for which partial memberships are obvious. It is clearly evident that, when both U and A are finite (as in the application references mentioned above) soft sets and fuzzy soft sets could be represented either by matrices or in tabular form. Rows are attached with objects in U , and columns are attached with parameters in A. In the case of a soft set, these representations are binary (i.e., all cells are either 0 or 1). The most successful approaches for fuzzy soft set based decision making have probably been done by Roy and Maji [23], Kong et al. [30], Feng et al. [31] and Alcantud [24].

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2.2. The need for a separate study of negative attributes To our knowledge, the question of negative attributes seems to have been neglected in the study of fuzzy soft set based decision making until Alcantud [32]. In this field, it is universally admitted that the higher the degree of an option to the set of F (e)-approximations, the better the option is, irrespective of the attribute e. However we are often interested in negative attributes of the alternatives. For example, in the widely-used problem of seeking a house we may be interested in knowing how safe its neighborhood is. We may then use crime databases, which provide an intrinsically negative measure. Alternatively, we may ask the neighbors. However, psychologists are aware that people are often unwilling to admit negative attitudes and beliefs about social aspects (cf., Fazio et al. [33]). So, we come across two difficult issues which are not matching: sometimes we want or have to use negative information (e.g., data on absenteeism or users complaints), and when it comes from subjective opinions, that information is not immediately transformable into positive scales (cf., Wong et al. [34]). In order to overcome these difficulties we have a flexible tool in the form of fuzzy complements (cf., Klir and Yuan [22, Sect. 3.2]). They provide a versatile way to transform negative into positive attributes with the possibility to account for the negativity bias. In the rest of this section we investigate to what extent fuzzy complements can interact with existing algorithms in order to enhance decision making operations with fuzzy soft sets. In relation to this aspect, the practitioner must take into consideration that attributes are not universally classifiable as positive or negative. There are attributes that can be positive for one decision maker but negative for another one. We explain this situation with the following example. 5

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Example 1. A specialized journal evaluates cars according to the attributes ‘length’ and ‘width’ in such a way that large values mean large size. Thus, for a reader who lives in the city center and has a small-sized parking slot, these attributes are negative. Of course, for other persons they may well be positive. If an agent is incorporating the journal’s information into her decision making process, and these attributes belong to her parameter class ‘dimensions’ which also contains attributes like ‘suitable for my family size’ or ‘enough number of seats’, then the agent needs to work with both positive and negative attributes in the same class.

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2.3. Fuzzy complements A fuzzy complement is a mapping c : [0, 1] −→ [0, 1] that satisfies at least the following two axiomatic requirements (cf., Klir and Yuan [22, Sect. 3.2]):

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i) Boundary condition: c(0) = 1 and c(1) = 0. ii) Monotonicity: for all a, b ∈ [0, 1], if a 6 b then c(a) > c(b).

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The specialized literature has frequently used the following three classes of fuzzy complements:

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1. Zadeh’s fuzzy complement: c(x) = 1 − x. 1−a , λ > −1. 1 − λa 1 3. Yager class of fuzzy complements: cω (x) = (1 − aω ) ω , ω > 0.

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2. Sugeno class of fuzzy complements: cλ (x) =

Sugeno’s complement with λ = 0 and Yager’s complement with ω = 1 coincide with Zadeh’s complement. These classes of complements satisfy another requirement that is desirable in most cases of practical significance: they are involutive, i.e., they verify c(c(a)) = a for all a ∈ [0, 1]. Involutive fuzzy complements are continuous and bijective (cf., [22, Th. 3.1]). However, there are continuous but not involutive fuzzy complements, like the Cosine fuzzy complement q(a) = 21 (1 + cos(πa)). In addition, there are fuzzy complements that are neither involutive nor continuous, like the threshold-type fuzzy complements. In the tradition of the field, throughout the rest of this paper we refer to the important Sugeno and Yager classes for comparison.

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2.4. On the role of fuzzy negations in fuzzy soft set based decision making We have shown that we can make a convincing case for the analysis of negative attributes or parameters in fuzzy soft set based decision making. Now, we intend to argue that this problem is not trivial, and deserves an exhaustive investigation. To that purpose in this paragraph we will be concerned with the consequences of the following assumption:

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Assumption 1. In order to apply a decision making (DM) procedure to a fuzzy soft set based choice situation with both positive and negative attributes, we first make the input data uniform by applying a given fuzzy complement to all elements associated with negative attributes. Then we apply that procedure to the uniform representation of the problem.

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The implementation of Assumption 1 depends on two factors: the fuzzy complement and the decision making procedure. Regarding fuzzy complements, the tradition in the field invites us to explore the cases in section 2.3. With respect to decision making, we are interested in the following two solutions. Roy and Maji [23] proposed to use the following algorithm:

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Roy and Maji’s algorithm.

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Step 1. Input a fuzzy soft set (F, A) on k objects as an input table whose cell (i, j) is denoted tij .

Step 2. Construct Roy and Maji’s Comparison matrix C = (cij )k×k where cij is the number of parameters for which tim − tjm > 0.

Step 3. For i = 1, ..., k, let ri be the sum of the elements in row i of C, and ti be the sum of the elements in column i of C. Compute scores si = ri − ti of objects i = 1, ..., k.

Step 4. The decision is any object ok that maximizes the score, i.e., any ok such that sk = maxi=1,...,k si .

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Alcantud [24] (see also Mathew and Alcantud [35]) proved that a computationally simple modification of the above algorithm made it possible to avoid ties to a large extent. Hence, the following alternative was endorsed:1

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Alcantud’s algorithm.

In Roy and Maji’s algorithm, replace Step 2 by the following:

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Step 2. For parameters j = 1, ..., q, let Mj be the maximum membership value of any object, i.e., Mj = maxi=1,...,k tij .

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Construct a Comparison matrix A = (aij )k×k where for each i, j, aij is the sum of the non-negative values in the finite sequence

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tiq − tjq ti1 − tj1 ti2 − tj2 , , ......, . M1 M2 Mq

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Alcantud [32] investigates variations of existing decision making procedures that incorporate the standard fuzzy complement. Therefore the paradigm in [32] was much more restrictive than our position here, and only an outline of its possibilities were given. With these preliminaries, let us show the application of Assumption 1 in an example:

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Example 2. We consider a set of cars U = {o1 , o2 , o3 , o4 }. They have attributes in A = {a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 }. The universal set of parameters E contains A. The problem is characterized by the fuzzy soft set defined by the tabular representation in Figure 1. An agent intends to use those data in order to make a decision. She regards attributes b1 , b2 , c1 and c2 as negative. We are bound by Assumption 1, and we select the standard fuzzy complement c(x) = 1 − x together with (1) Alcantud’s algorithm, and (2) Roy and Maji’s algorithm. 1

Both papers [23, 24] work with multi-parameter fuzzy soft sets. With them a resultant fuzzy soft set is produced, and then these algorithms are applied. Hence the proposed algorithms are richer than we show here (cf., [24, Table 13]). Nevertheless, for the purpose of illustration we prefer to streamline the argument.

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0.3 0.4 0.6 0.4 0.2 0.8 0.3 0.4 0.1 0.3 0.9 0.3 0.8 0.6 0.3 0.6 0.5 0.4 0.4 0.5 0.8 0.6 0.4 0.4 0.5 0.6 0.3 0.8 0.2 0.4 0.9 0.8 0.2 0.7 0.6 0.6

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Therefore, we first need to make the tabular representation of the fuzzy soft set in Figure 1 uniform. Figure 2 gives that uniform tabular representation of (F, A) for the agent.

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0.3 0.4 0.6 0.6 0.8 0.8 0.7 0.6 0.1 0.3 0.9 0.3 0.2 0.4 0.3 0.4 0.5 0.4 0.4 0.5 0.8 0.4 0.6 0.4 0.5 0.4 0.3 0.8 0.2 0.4 0.1 0.2 0.2 0.3 0.4 0.6

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Figure 1: Tabular representation of the fuzzy set (F, A) associated with Example 2.

Figure 2: Uniform tabular representation of the fuzzy set (F, A) associated with Example 2 and the agent’s selection of negative attributes, with the standard fuzzy negation.

(1) From the uniform tabular representation (F, A) in Figure 2, it is easy to compute its Comparison table by Alcantud’s algorithm (cf., [24]), which is given in Table 1. Then Table 2 shows its associated scores, which suggest the ordering o1  o3  o2  o4 . As a result, one concludes that o1 should be selected when we consider the input data of Example 2. (2) If we apply Roy and Maji’s algorithm to the uniform tabular representation (F, A) in Figure 2, then the corresponding Comparison matrix is shown in Table 3. Table 4 shows its associated scores. Therefore we obtain the same conclusion o1  o3  o2  o4 , and the advice is that o1 should be selected when we consider the input data of Example 2. 9

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0 2.76 1.7 3.71 1.06 0 0.78 1.63 0.82 1.6 0 2.37 1.46 1.08 1 0

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8.17 3.47 4.79 3.54

3.34 5.44 3.48 7.71

Score (Si )

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4.83 −1.97 1.31 −4.17

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Table 1: Comparison table of the fuzzy soft set in Example 2 by Alcantud’s algorithm.

Table 2: Score table of the fuzzy soft set in Example 2 by Alcantud’s algorithm.

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Table 3: Comparison table of the fuzzy soft set in Example 2 by Roy and Maji’s algorithm.

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Row-sum (Ri )

Column-sum (Ti )

19 12 17 8

9 16 11 20

Score (Si ) 10 −4 6 −12

Table 4: Score table of the fuzzy soft set in Example 2 by Roy and Maji’s algorithm.

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In Example 2 we have implemented Assumption 1 in a case where there are both positive and negative parameters. The gist of the example is that in order to make the data uniform, the standard Fuzzy complement is computed for the negative attributes prior to the application of the corresponding Algorithm. However, one could have also proceeded with the application of the same decision making mechanisms to the uniform table obtained from the calculation of other fuzzy complements for the negative attributes. The question arises: does the choice of the fuzzy complement alter the result (the other factors remaining unchanged)? Our natural question motivates the following novel notions:

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Definition 3. A fuzzy soft set based decision procedure is insensitive to the choice of a fuzzy complement if for any fixed problem, the application of Assumption 1 with such a procedure always produces the same ordering outcome, irrespective of the fuzzy complement. Otherwise we say that the procedure is sensitive to the choice of a fuzzy complement.

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In Proposition 1 below we prove that the notions in Definition 3 are nontrivial, i.e., that there are procedures that are insensitive and procedures that are sensitive to the choice of a fuzzy complement:

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Proposition 1. Roy and Maji’s algorithm is insensitive to the choice of a fuzzy complement. However, Alcantud’s algorithm is sensitive to the choice of a fuzzy complement. Proof. The first statement is due to monotonicity of the fuzzy complements. It ensures that Roy and Maji’s Comparison matrix is the same for any fuzzy complement that we select. Therefore the si scores in Roy and Maji’s algorithm do not change when we vary the fuzzy complement. The second statement is proven by examples. It suffices to check that in the situation of Example 2, we obtain a different ordering solution when we use Yager’s fuzzy complement with adequate parameters. We perform an exhaustive analysis in Example 3 below, which proves our claim.  Example 3. We proceed to experiment on the input data of Example 2 for application with Alcantud’s algorithm (cf., [24]) and Yager’s class of fuzzy 11

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complements. We observe that the changes in the ordering are dramatic in this case. If we start off with a small value of ω = 0.001, it produces an ordering of o3  o4  o2  o1 . This ordering is altered for a value of ω = 0.007 and the new ordering is o1  o3  o4  o2 . For ω = 0.1, Table 5 shows the Comparison table, and the Score table is shown in Table 6. As the ω value is altered to 0.167, a new ordering of o1  o3  o2  o4 is obtained. This ordering is maintained until ω = 2.2. A new ordering of o3  o1  o2  o4 is generated at ω = 2.3. The Comparison table and Score table for ω = 0.4 are shown in Tables 7 and 8, respectively. This is similar to the results we obtained in Example 2 under its specific requirements. At ω = 5, the new ordering is as o3  o1  o4  o2 . The Comparison table for ω = 5.4 is shown in Table 9, and the corresponding Score table appears in Table 10, which suggests the same ordering as o3  o1  o4  o2 . For an ω value of 5.5 , a new ordering of o3  o4  o1  o2 is obtained and this ordering continues until ω = 6.3. As ω again increases, the previous ordering gets renewed at ω = 6.4 and Table 11 shows the Comparison table for ω = 10. Table 12 shows its associated scores, which suggest the ordering o3  o4  o2  o1 . This ordering remains stable for further higher values of ω. o1

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0 4.93 4.48 5.22 1.06 0 0.68 0.97 0.82 0.89 0 1.1 1.46 1.08 1 0

Table 5: Comparison table of the fuzzy soft set in Example 3 for ω = 0.1.

Table 13 summarizes the ordering solutions for Example 2 under various fuzzy complements and the two selected algorithms, when Assumption 1 is imposed in order to deal with negative attributes. The Comparison matrices and Score tables for the Sugeno class of complements can be computed on the same lines as shown for Yager complements, hence we skip them. Its last line captures the fact that Roy and Maji’s algorithm is insensitive to the choice of a fuzzy complement. 12

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Column-sum (Ti )

14.63 2.71 2.81 3.54

3.34 6.9 6.16 7.29

Score (Si )

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11.29 −4.19 −3.35 −3.75

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0 4.15 3.28 4.85 1.06 0 0.88 1.38 0.82 1.51 0 1.93 1.46 1.08 1 0

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Table 6: Score table of the fuzzy soft set in Example 3 for ω = 0.1.

Table 7: Comparison table of the fuzzy soft set in Example 3 for ω = 0.4.

12.28 3.32 4.26 3.54

3.34 6.74 5.16 8.16

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Score (Si ) 8.94 −3.42 −0.9 −4.62

Table 8: Score table of the fuzzy soft set in Example 3 for ω = 0.4.

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0 1.09 0.53 1.47 1.06 0 0.62 1.06 0.82 0.95 0 1.3 1.46 1.08 1 0

Table 9: Comparison table of the fuzzy soft set in Example 3 for ω = 5.4.

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Column-sum (Ti )

3.09 2.74 3.07 3.54

3.34 3.12 2.15 3.83

Score (Si )

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−0.25 −0.38 0.92 −0.29

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Row-sum (Ri )

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0 1.01 0.5 1.28 1.06 0 0.61 0.95 0.82 0.89 0 1.14 1.46 1.08 1 0

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Table 10: Score table of the fuzzy soft set in Example 3 for ω = 5.4.

Table 11: Comparison table of the fuzzy soft set in Example 3 for ω = 10.

2.79 2.62 2.85 3.54

3.34 2.98 2.11 3.37

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Score (Si ) −0.55 −0.36 0.74 0.17

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Table 12: Score table of the fuzzy soft set in Example 3 for ω = 10.

3. An algorithm for decision making under separable fuzzy soft set In this section we investigate decision making in the following standard but novel situation. Let U = {o1 , o2 , . . . , ok } be a set of k objects. It may be characterised by a set of parameters. These parameters are classified as A1 , A2 , . . . , Aq , i.e, each subset Aj represents the j-th class of parameters. The elements of Aj are disjoint. It is assumed that these property sets may be viewed as fuzzy sets. The parameter space E may be any set with the property A1 ∪A2 ∪. . .∪Ai ∪{A1 }∪. . .∪{Aq } ⊆ E. We explain the advantage of this requirement below in section 3.2. 14

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Standard Yager

Alcantud Alcantud

Sugeno

Alcantud

Any one

Roy and Maji

Irrelevant [0, 0.006) [0.007, 0.166) [0.167, 2.2) [2.3, 4.9) [5, 5.4) [5.5, 6.3) [6.4, +∞) [-0.999, -0.991) [-0.990, -0.983) [-0.982, -0.975) [-0.974, -0.859) [-0.858, +∞ ) Irrelevant

Prescribed ordering o1 o3 o1 o1 o3 o3 o3 o3 o3 o3 o3 o3 o1 o1

 o3  o4  o3  o3  o1  o1  o4  o4  o4  o4  o1  o1  o3  o3

 o4  o1  o2  o4  o4  o2  o2  o1  o1  o2  o2  o4  o4  o4

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Parameter interval

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DM procedure

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Fuzzy compl.

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Table 13: Ordering solutions for Example 2 under various fuzzy complements and focal algorithms. We are bound by Assumption 1.

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The idea is that when we can categorize the parameters by attribute classes, we can perform more accurate evaluations of the alternatives. Now we proceed to formalize our novel concept of a separable fuzzy soft set, and then we propose a flexible decision making procedure for that setting. 3.1. Separable fuzzy soft set The formal definition of the concept under investigation is as follows: Definition 4. A separable fuzzy soft set on U is a triple (F, A, {A1 , A2 , . . . , Aq }) where A1 , A2 , . . . , Aq is a partition of A. 2 Under finiteness, separable fuzzy soft sets can be presented in tabular form, where the parameters are grouped as prescribed by the partition. Informally, the separable fuzzy soft set (F, A, {A1 , A2 , . . . , Aq }) on U can also 2

This means that A1 ∪ A2 ∪ . . . ∪ Aq = A, Aj ∩ Al = ∅ for each j, l, and Aj 6= ∅ for each j.

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be described by q fuzzy soft sets (F, A1 ), . . . , (F, Aq ) on U . Let us see these possibilities with an example:

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Example 4. We consider a set of cars U = {o1 , o2 , o3 , o4 }. The decision maker is interested in the attributes A = {fashionable (a1 ), visible on road (a2 ), easy to keep clean (a3 ), large (b1 ), wide (b2 ), enough number of seats (b3 ), gross price (c1 ), taxes (c2 ), subsidies for being environmentally friendly (c3 )}. A separable fuzzy soft set (F, A, {A1 , A2 , A3 }) on U is defined by the tabular form in Figure 3, where (a) a colour space is represented by A1 = {a1 , a2 , a3 }, (b) a size space is represented by A2 = {b1 , b2 , b3 }, (c) a price space is represented by A3 = {c1 , c2 , c3 }. We can describe the same situation with the following convention. A fuzzy soft set (F1 , A1 ) describes the ‘objects having colour space’. A fuzzy soft set (F2 , A2 ) describes the ‘objects having size’. A fuzzy soft set (F3 , A3 ) describes the ‘objects having price’. They are defined by the tabular representations in Figure 4. These three fuzzy soft sets naturally define a unique separable fuzzy soft set on A, which has been described above in this example. (F, A) 0.3 0.3 0.4 0.8

0.4 0.9 0.5 0.2

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A1

0.6 0.3 0.8 0.4

A2

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b1

b2

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c1

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0.4 0.8 0.6 0.9

0.2 0.6 0.4 0.8

0.8 0.3 0.4 0.2

0.3 0.6 0.5 0.7

0.4 0.5 0.6 0.6

0.1 0.4 0.3 0.6

Figure 3: Tabular representation of the separable fuzzy soft set (F, A, {A1 , A2 , A3 }) in Example 4.

It is important to stress that separable fuzzy soft sets and multi-parameter fuzzy soft sets (Roy and Maji [23], Alcantud [24]) are designed to capture different choice circumstances. The reader should recall that the second case refers to situations where the decision maker assesses the alternatives through certain combinations of attributes. Here we do not mix up parameters from different classes. We combine them in order to produce a resultant fuzzy soft set too, but instead we proceed by attribute classes as the next section shows. 16

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(F2 , B) 0.3 0.4 0.3 0.9 0.4 0.5 0.8 0.2

a3 0.6 0.3 0.8 0.4

o1 o2 o3 o4

(F3 , C) c1

c3

b2

b3

0.4 0.8 0.6 0.9

0.2 0.6 0.4 0.8

0.8 0.3 0.4 0.2

0.3 0.4 0.1 0.6 0.5 0.4 0.5 0.6 0.3 0.7 0.6 0.6

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(F1 , A)

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Figure 4: Tabular representations of the auxiliary fuzzy soft sets (F1 , A1 ), (F2 , A2 ) and (F3 , A3 ) in Example 4.

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3.2. From separable fuzzy soft sets to H-resultant fuzzy soft sets Extended aggregation operators (cf., Beliakov et al. [36, Def. 1.6]) allow us to define H-resultant fuzzy soft sets associated with separable fuzzy soft sets and the extended aggregation operator H, in the following terms. An extended aggregation operator is a mapping [ H: [0, 1]n −→ [0, 1] n=1,2,...

that verifies (i) H(0, . n. ., 0) = 0, H(1, . n. ., 1) = 1, for each n = 1, 2, . . . (ii) x 6 y implies H(x) 6 H(y), for all x, y ∈ [0, 1]n . (iii) H(a) = a when a ∈ [0, 1]. Henceforth, we assume that A, the set of parameters is finite. This ast(1) t(q) sumption allows us to decompose A = {a11 , . . . , a1 , . . . , a1q , . . . , aq } where t(j) for each j = 1, . . . , q, Aj = {a1j , . . . , aj } ⊆ A. For each j = 1, . . . , q, we fix a label associated with the class parameter j. We denote such label by Aj . In formal set-theoretic terms, we might 17

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use Aj = {Aj }, the set whose only element is the set Aj , which in turn has t(j) elements (parameters). In this fashion we justify the original requirement that A1 ∪A2 ∪. . .∪Ai ∪{A1 }∪. . .∪{Aq } ⊆ E. This ensures that the universal set of attributes also contains the new labels for our class parameters.

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FH : A = {A1 , . . . , Aq } −→ FS(U )

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Definition 5. Let H be any extended aggregation function. For any separable fuzzy soft set (F, A, {A1 , A2 , . . . , Aq }) on U , the H-resultant fuzzy soft set (FH , {A1 , A2 , . . . , Aq }) on U is defined by the expression

an

such that that for each j, the fuzzy set FH (Aj ) : U −→ [0, 1] is defined as t(j) FH (Aj )(ol ) = H(F (a1j )(ol ), . . . , F (aj )(ol )) for every ol ∈ U .

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Now we show the application of the notion of H-resultant fuzzy soft set associated with a separable fuzzy soft set in an example.

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Example 5. We assume the setting of Example 4, and fix H as the arithmetic mean aggregation function. The H-resultant fuzzy soft set (FH , {A1 , A2 , A3 }) on U is defined by the tabular form in Figure 5, where A1 is a label for the parameter class ‘colour’, A2 is a label for the parameter class ‘size’, and A3 is a label for the parameter class ‘price’. We abbreviate A = {A1 , A2 , A3 } for convenience.

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(FH , A) o1 o2 o3 o4

A1

A2

A3

1.3 3

1.4 3 1.7 3 1.4 3 1.9 3

0.8 3

0.5 1.7 3 1.4 3

0.5 1.4 3 1.9 3

Figure 5: Tabular representation of the H-resultant fuzzy soft set (FH , A) in Example 5.

The numbers in the tabular representation given by Figure 5 are computed as follows. The fuzzy set FH (A1 ) : U −→ [0, 1] is represented by the first column, as is standard. Now because A1 = {a1 , a2 , a3 } we compute FH (A1 )(o1 ) = H(F (a1 )(o1 ), F (a2 )(o1 ), F (a3 )(o1 )) = 0.3+0.4+0.6 = 1.3 . The 3 3 rest of the cells are computed in a similar manner. 18

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3.3. Separable fuzzy soft set and its decision making Now we are in a position to propose the following flexible mechanism for making decisions about alternatives that are characterized by separable fuzzy soft sets:

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Algorithm 1 - A priori factors: selection of all negative attributes, choice of the fuzzy complement (e.g., Zadeh’s, Sugeno or Yager classes of fuzzy complements with a fixed parameter), and choice of the interclass aggregation function H (e.g., arithmetic or geometric mean).

1: Input the separable fuzzy soft set (F, A, {A1 , A2 , . . . , Aq }) on U in tabular

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form. The input can also be represented by (F1 , A1 ), . . . , (Fq , Aq ), as in Example 4 2: For each negative parameter j, replace column j by the result of applying the selected fuzzy complement to each of its cells. We can represent this uniformed separable fuzzy soft set by the uniformed fuzzy soft sets (F10 , A1 ), . . . , (Fq0 , Aq ) in tabular form. 3: Compute one H-resultant fuzzy soft set on k objects o1 , . . . , ok using the selected interclass aggregation operator on the uniformed data, and place it in the form of a k × q table T whose cell (i, j) is denoted Tij . Column i is associated with the corresponding uniformed fuzzy soft set (Fi0 , Ai ) given in step 2. 4: Consider table T . For each parameter j, let Mj be the maximum column value of any object, i.e., Mj = maxi=1,...,k tij for each j = 1, ..., q. Now construct a k × k comparison matrix A = (aij )k×k where for each i, j, we let aij be the sum of the non-negative values in the following finite sequence: tiq − tjq ti1 − tj1 ti2 − tj2 , , ......, . M1 M2 Mq

A can be identified with a comparison table as well. 5: For each i = 1, ..., k, compute Ri as the sum of the elements in row i of A, and Ti as the sum of the elements in column i of A. Then for each i = 1, ..., k, compute the score Si = Ri − Ti of object i. 6: The decision is any object ok that maximizes the score, i.e., any ok such that Sk = maxi=1,...,k Si .

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The following example illustrates the application of Algorithm 1 to a concrete situation, namely, the separable fuzzy soft set in Example 4 above.

(F20 , B)

(F10 , A) 0.3 0.4 0.3 0.9 0.4 0.5 0.8 0.2

a3

b1

0.6 0.3 0.8 0.4

o1 o2 o3 o4

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(F30 , C)

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Example 6. Let us assume that we have the input data of Example 4. An agent intends to use those data in order to make a decision. She regards attributes b1 , b2 , c1 and c2 as negative, and she √ decides to use a Yager’s complement with factor 0.5, i.e., c0.5 (a) = (1 − a)2 . As an interclass aggregation function, the geometric mean is selected. Figure 6 gives the uniformed fuzzy soft sets obtained in Step 2 of Algorithm 1.

c1

b3

0.135 0.306 0.8 0.011 0.051 0.3 0.051 0.135 0.4 0.003 0.011 0.2

c3

0.205 0.135 0.1 0.051 0.086 0.4 0.086 0.051 0.3 0.027 0.051 0.6

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o1 o2 o3 o4

c2

b2

Figure 6: Tabular representations of the uniformed fuzzy soft sets (F10 , A), (F20 , B) and (F30 , C) associated with Example 4.

Table 14 gives the H-resultant fuzzy soft set obtained in Step 3 of Algorithm 1. Now it is easy to compute its Comparison table by Algorithm 1, which is given in Table 15. Then Table 16 shows its associated scores. The solution is o1  o3  o2  o4 and one concludes that o1 should be selected when we consider the input data of Example 4. 3.4. Remarks on the distinctive characteristics of our proposal To end up this section, we offer some remarks that explain important issues. 20

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price

0.416 0.321 0.433 0.055 0.543 0.140 0.4 0.018

0.140 0.120 0.109 0.093

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o1 o2 o3 o4

size

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o2

o4

0 0.031 0.234 0

0.97 0.784 1.308 0 0.078 0.369 0.467 0 0.757 0 0 0

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o1

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o1 o2 o3 o4

o3

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Table 14: The tabular representation T of the H-resultant fuzzy soft set using Algorithm 1 in Example 4.

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Table 15: Comparison table that represents the comparison matrix A of the H-resultant fuzzy soft set in Example 4 using Algorithm 1.

3.062 0.478 1.458 0

0.265 1.437 0.862 2.434

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Column-sum (Ti )

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o1 o2 o3 o4

Row-sum (Ri )

Score (Si ) 2.797 −0.959 0.596 −2.434

Table 16: Score table of the resultant fuzzy soft set given in Table 15.

(i) In Algorithm 1, we can dispense with fuzzy complements if all the parameters are positive. And we do not need to select H if every parameter is a separate class, as is the case with standard fuzzy soft sets. (ii) In the parameterization of the alternatives it may happen that a class of parameters has both positive and negative parameters, whereas other classes have only positive (respectively, negative) parameters. For this reason one should not aggregate the raw information by a common interclass aggregation function. Actually, the aggregation of such information would 21

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not be simple at all. In order to neutralize this inconvenience we propose to make the information uniform at Step 2 so that then we can safely aggregate it at Step 3. (iii) In the parameterization of the alternatives by extended soft set models like fuzzy soft sets, it may happen that a class of attributes (for example, “engine” in the description of a car) has many more attributes than other classes (like “colour”, “size” or “prize”). Many decision making algorithms (for example, the widely accepted choice value criterion for soft sets) overrate classes with a larger number of attributes. In order to compensate for a factor that may be completely anecdotal, the use of an aggregator is an alternative to weights (like weighted fuzzy soft sets in Feng et al. [31] or weighted fuzzy soft multisets in Das [10]). Instead of deciding which weights should be associated with each of the many attributes in the problem, we propose that one single decision suffices – that is, the interclass aggregation function. To sum up, our methodology takes advantage of the fact that aggregating and then making a decision produces a solution that is in general different from the solution when we make the decision directly from the original data. This feature can compensate for the excess of information about one or several kinds of attributes. The case study in section 3.5 illustrates this point in a real situation. The class of parameters “personal teaching abilities” is surely overrated and we exert an opposite influence by distributing the parameters into suitable classes. (iv) As is customary, the flexibility of the proposed algorithm comes at the cost of sensitivity to the choice of the factors. Put differently, the selection of the suitable fuzzy complement and aggregation function H should in general affect the final decision, since this is the justification of adjustability. With respect to the first issue, Klir and Yuan [22, page 50] explain that “involutive fuzzy complements play by far the most important role in practical applications”. To be more specific, Vreeswijk [37, Section 7.1] recalls the arguments that support the choice of the standard fuzzy negation under lack of additional arguments (cf., [38]). Fuzzy complements may be related to attitudes towards risk (Simonovi´c [39], Choudhary and Raghuvanshi [40]). They are at the core of the debate on opposition-based models in knowledge representation (Rodr´ıguez et al. [41]). As to the second issue, the determination of suitable aggregators has been the subject of interesting analyses in the specialized literature. We suggest the reader consult Beliakov and Warren [42], Beliakov [43] and Luo and Jennings [44], who have contributed 22

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to this debate with arguments from different perspectives.

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3.5. An application oriented case study In order to show the significance of our study, we now present a real case study with some characteristics that support the discussion in the previous section 3.4. In this Section, an application of our proposed methodology in teachers’ performance rating is shown. These rating surveys are often the only way to rank teaching ability based on the feedback of students and it helps teachers to become more aware of their own teaching practices. Intermittently, these are used for other practical purposes such as compensations and benefits. We explain our procedure with the real data collected from student evaluations of teaching staff at the School of Economics and Business, Universidad de Salamanca, Spain. Table 17 contains this student data, which has been made anonymous. The students of this University have to provide biyearly feed-back on specific areas of their learning experiences for each subject and teacher. The students record their confidential ranking level (0 means “strongly disagree”, 5 means “fully agree”) to statements such as “The teacher explains clearly”, “The teacher clears up any doubts raised in the classroom”, “Our activities and tasks are well organized and structured”, or “Overall, my degree of satisfaction with him/her has been good”. The student evaluations are then aggregated by subjects and are confidentially submitted to the teacher evaluated. Figure 7 illustrates an anonymized part of a subject report of an evaluated person, which is not used in our study. The sample questionnaire has four main columns, the first column of which is the only part devoted to the individual teacher (the other columns are provided for comparison: average of the Department and of the Degree Program). In Table 17, each row corresponds to one evaluated person in our sample, while each column coincides with the eleven fields in the survey (attributes). These figures have been normalized to the range [0, 1] by dividing them by 5. These normalized assessments now easily fit in as a separable fuzzy soft set. The items 1 to 7 in the survey represent a class A1 - personal teaching abilities, items 8 and 9 concern a class A2 - suitability of didactic resources, item 10 constitutes a class A3 - evaluation strategies and item 11 constitutes a class A4 - overall satisfaction. In our view, amends should be made for the excess information in class A1 as clarified in Section 3.4 (iii). Our proposal works out aggregation over weights to compensate for this.

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ENCUESTA DE SATISFACCIÓN DE LOS ESTUDIANTES CON LA ACTIVIDAD DOCENTE DEL PROFESOR

INFORME CONFIDENCIAL CURSO ACADÉMICO 2015-2016

UNIDAD DE EVALUACIÓN DE LA CALIDAD

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Carlos PROFESOR: We 29168946-R apply Algorithm 1 onRodríguez theAlcantud, staffJoséperformance ranking data for the Economía e Historia Económica Departamento: academic year 2015-16, which Fundamentos is shown in Table 17. We refrain from insisting del Análisis Económico Área de Conocimiento: on the use of fuzzy negation as all parameters are positive. The geometric 303649 NIVELACIÓN EN ESTADÍSTICA Y MATEMÁTICAS / Grupo: 1 ASIGNATURA / GRUPO: mean was selected as an interclass aggregation function and we proceeded PRIMER CUATRIMESTRE Periodo docente: using the steps Titulación: in Example 6 MÁSTER through Tables 18 (H-resultant fuzzy soft set) U. EN INVESTIGACIÓN EN ADMINISTRACIÓN Y ECONOMÍA DE LA EMPRESA Facultad de Economía y Empresa and 19 (Comparison table). It concludes in Table 20 with the Score Table Centro: Campus "Miguel de Unamuno" Campus: that suggests the ordering o2  o6  o5  o1  o3  o4 . Ciencias Sociales y Jurídicas

Rama:

Media de los estudiantes que responden la encuesta: Responden a la encuesta:

4

/

Matriculados en la asignatura / grupo:

7

=

Porcentaje de respuesta:

57.143

%

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DATOS DE CONTEXTO

Número de convocatoria:

1.00

Asistencia a clase (1)

4.00

CUESTIONARIO Valoraciones medias de las respuestas de los estudiantes(2):

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(1) Igual o inferior a 1 (25% o menos) Superior a 1 e igual o inferior a 2 (Entre 26-50%) Superior a 2 e igual o inferior a 3 (Entre 51-75%) Superior a 3 e igual o inferior a 4 (76% o más)

Media del Profesor

No procede(3)

Media Departamento(4)

Media Titulación(4)

3.25

0

3.44

4.24

El/la profesor/a explica con claridad.

2

Resuelve las dudas plateadas y orienta a los estudiantes en el desarrollo de sus tareas.

4.00

0

3.58

4.37

3

Organiza y estructura bien las actividades o tareas que realizamos con el/la profesor/a (aula, laboratorio, taller, seminario, trabajo de campo, etc.).

4.00

0

3.51

4.23

4

Las actividades o tareas (teóricas, prácticas, de trabajo individual, en grupo, etc.) son provechosas para lograr los objetivos de la asignatura.

4.00

0

3.46

4.23

5

Favorece la participación del estudiante en el desarrollo de la actividad docente.

3.25

0

3.37

4.30

6

Está accesible para ser consultado por los estudiantes (tutorías, orientación académica, ...)

3.75

0

3.76

4.44

7

Ha facilitado mi aprendizaje y gracias a su ayuda he logrado mejorar mis conocimientos, habilidades o destrezas.

3.25

0

3.26

4.28

8

Los recursos didácticos utilizados por el/la profesor/a son adecuados para facilitar el aprendizaje.

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1

0

3.34

4.21

4.50

0

3.40

4.31

Los métodos de evaluación se corresponden con el desarrollo docente de la 10 materia (responder sólo en caso de que se hayan realizado pruebas de evaluación de la asignatura).

4.25

0

3.47

4.28

11 Mi grado de satisfacción general con el/la profesor/a ha sido bueno.

3.75

0

3.43

4.34

3.46

4.29

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4.25

La bibliografía y los materiales didácticos facilitados son útiles para realizar las tareas y para el aprendizaje.

9

(2) 1 = Totalmente en desacuerdo, 2 = En desacuerdo, 3 = Ni acuerdo ni desacuerdo, 4 = De acuerdo, 5 = Totalmente de acuerdo (3) Incluye la valoración en número de los estudiantes que marcan “no procede” para cada una de las preguntas. (4) Medias referidas a la actividad docente desarrollada en este curso académico

Media:

3.84

Figure 7: Anonymized part of a subject report, Universidad de Salamanca (in Spanish). Unidad de Evaluación de la Calidad - Calle Traviesa 3-7, 2º. 37008 - Salamanca - E-mail: qualitas@ usal.es - Web: http://qualitas.usal.es - Tel./Fax: (34) 923 29 46 38

4. Conclusions In this contribution we have shown that the study of fuzzy soft set based decision making with positive and negative attributes deserves careful consideration. 24

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A1

A2

A3

1

2

3

4

5

6

7

8

9

10

s1 s2 s3 s4 s5 s6

0.46 0.75 0.51 0.48 0.78 0.75

0.58 0.79 0.53 0.57 0.78 0.80

0.53 0.69 0.64 0.46 0.54 0.62

0.62 0.68 0.64 0.61 0.67 0.64

0.49 0.78 0.58 0.48 0.78 0.77

0.71 0.67 0.68 0.60 0.78 0.76

0.56 0.74 0.64 0.46 0.78 0.77

0.58 0.67 0.71 0.63 0.72 0.72

0.64 0.78 0.67 0.66 0.67 0.74

0.78 0.62 0.71 0.63 0.62 0.64

11

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A4

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0.68 0.81 0.60 0.61 0.66 0.73

0.56 0.73 0.60 0.52 0.72 0.73

0.61 0.72 0.69 0.64 0.69 0.73

A3

A4

0.78 0.62 0.71 0.63 0.62 0.64

0.68 0.81 0.60 0.61 0.66 0.73

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s1 s2 s3 s4 s5 s6

A1

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Table 17: A representation of the separable fuzzy soft set associated with the case study in section 3.5. Class A1 is “personal teaching abilities”, A2 is “suitability of didactic resources”, A3 is “evaluation strategies” and A4 is “overall satisfaction”.

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Table 18: The tabular representation of the H-resultant fuzzy soft set using Algorithm 1 in Example 4. H is the geometric mean.

o1 o2 o3 o4 o5 o6

o1

o2

o3

o4

0 0.547 0.166 0.049 0.344 0.458

0.205 0.189 0.334 0 0.48 0.64 0.115 0 0.275 0.013 0.012 0 0 0.252 0.412 0.035 0.39 0.563

o5

o6

0.23 0.179 0.228 0.099 0.115 0.09 0.013 0 0 0 0.164 0

Table 19: Comparison table that represents the comparison matrix A of the H-resultant fuzzy soft set in section 3.5 using Algorithm 1.

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Column-sum (Ti )

1.137 1.994 0.761 0.087 1.008 1.61

1.564 0.368 1.323 2.224 0.75 0.368

Score (Si )

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−0.427 1.626 −0.562 −2.137 0.258 1.242

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o1 o2 o3 o4 o5 o6

Row-sum (Ri )

Table 20: Score table of the resultant fuzzy soft set given in Table 18.

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We have defined the notion of Sensitivity to the selection of a fuzzy complement. This is a criterion that discriminates among procedures for fuzzy soft set based decision making. We concluded that Roy and Maji’s [23] algorithm is insensitive, whereas Alcantud’s [24] algorithm is sensitive, to the choice of a fuzzy complement. Table 13 summarizes our comparison between the new DM procedures that arise from the interaction with fuzzy complements. As future research lines, we believe that the above analysis can be extended to other methodologies, and also to related soft computing models such as Hesitant fuzzy linguistic terms sets (cf., Rodr´ıguez et al. [45, 46, 47], Dong et al. [48]). Finally, we have defined the new concept of separable fuzzy soft set, which also deserves additional studies. A novel flexible decision making procedure for alternatives characterized by such a concept has been proposed, and we have validated our model with a real case study. Acknowledgements

The authors thank the four anonymous reviewers, Prof. Bas van Vlijmen (managing editor), and Prof. Ren-Jieh James Kuo (handling editor) for their valuable comments and recommendations. The second author has contributed to this research while preparing his Ph.D. Dissertation at Mahatma Gandhi University.

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References

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[1] L. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.

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[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems 20 (1986) 87–96.

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[3] Z. Pawlak, Rough sets, International Journal of Computer & Information Sciences 11 (5) (1982) 341–356. [4] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems 25(6) (2010) 529–539.

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[5] J. C. R. Alcantud, R. de Andr´es Calle, M. J. M. Torrecillas, Hesitant fuzzy worth: An innovative ranking methodology for hesitant fuzzy subsets, Applied Soft Computing 38 (2016) 232–243.

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[6] D. Molodtsov, Soft set theory - first results, Computers and Mathematics with Applications 37 (1999) 19–31.

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[7] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Computers & Mathematics with Applications 57 (9) (2009) 1547–1553.

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[8] F. Feng, Y. Li, Soft subsets and soft product operations, Information Sciences 232 (2013) 44–57. [9] M. Ali, A note on soft sets, rough soft sets and fuzzy soft sets, Applied Soft Computing 11 (2011) 3329–3332.

[10] A. K. Das, Weighted fuzzy soft multiset and decision-making, International Journal of Machine Learning and Cybernetics (2016) forthcoming,doi:10.1007/s13042-016-0607-y. [11] F. Feng, Soft rough sets applied to multicriteria group decision making, Annals of Fuzzy Mathematics and Informatics 2 (1) (2011) 69–80. [12] F. Feng, C. Li, B. Davvaz, M. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing 14 (9) (2010) 899–911.

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[13] F. Feng, X. Liu, V. Leoreanu-Fotea, Y. B. Jun, Soft sets and soft rough sets, Information Sciences 181 (6) (2011) 1125 – 1137.

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[14] F. Hakim, E. Saari, T. Herawan, Soft solution of soft set theory for recommendation in decision making, in: T. H. et al. (Ed.), Recent Advances on Soft Computing and Data Mining, no. 287 in Advances in Intelligent Systems and Computing, Springer International Publishing, Switzerland, 2014, pp. 313–324.

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